Modularity and integral points on moduli schemes
The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems (e.g. -unit and Mordell equations), this gives an effective method which does not rely on Diophantine approximation or transcendence techniques. We also combine Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory, to establish the effective Shafarevich conjecture for abelian varieties of (product) -type. In particular, we open the way for the effective study of integral points on certain higher dimensional moduli schemes.
The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example -unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques. We also combine Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory, to establish the effective Shafarevich conjecture for abelian varieties of (product) -type. In particular, we open the way for the effective study of Diophantine equations related to integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties. In what follows in the introduction, we describe in more detail the content of this paper.
1.1 Integral points on moduli schemes of elliptic curves
To provide some motivation for the study of integral points on moduli schemes of elliptic curves, we discuss in the following section fundamental Diophantine equations which are related to such moduli schemes. For any , we denote by the usual (absolute) logarithmic Weil height of defined for example in [BG06, p.16].
1.1.1 -unit and Mordell equations
Let be a finite set of rational prime numbers. We define if is empty and with the product taken over all otherwise. Let denote the units of . First, we consider the classical -unit equation
The study of -unit equations has a long tradition and it is known that many important Diophantine problems are encapsulated in the solutions of (1.1). For example, any upper bound for which is linear in terms of is equivalent to a version of the -conjecture. Mahler [Mah33], Faltings [Fal83b] and Kim [Kim05] proved finiteness of (1.1) by completely different methods. Moreover, Baker’s method [Bak68c] or a method of Bombieri [Bom93] both allow in principle to find all solutions of any -unit equation. We will briefly discuss the methods of Baker, Bombieri, Faltings, Kim and Mahler in Section 7.2.1. In addition, we now point out that Frey remarked in [Fre97, p.544] that the Shimura-Taniyama conjecture implies finiteness of (1.1). It turns out that one can make Frey’s remark in [Fre97] effective and one obtains for example the following explicit result (see Corollary 7.2): Any solution of the -unit equation satisfies
(After we uploaded the present paper to the arXiv, Hector Pasten informed us about his joint work with Ram Murty [MP13] in which they independently obtain a (slightly) better version of the displayed height bound (see [MP13, Theorem 1.1]) by using a similar method; we refer to the comments below Corollary 7.2 for more details. We would like to thank Hector Pasten for informing us about [MP13].) Frey uses inter alia his construction of Frey curves. This construction is without doubt brilliant, but rather ad hoc and thus works only in quite specific situations. The starting point for our generalizations are the following two observations: The solutions of (1.1) correspond to integral points on the moduli scheme , and the construction of Frey curves may be viewed as an explicit Paršin construction induced by forgetting the level structure on the elliptic curves parametrized by the points of .
We now discuss a second fundamental Diophantine equation which is related to integral points on moduli schemes. For any nonzero , one obtains a Mordell equation
We shall see in Section 7.3 that this Diophantine equation is a priori more difficult than (1.1). In fact the resolution of (1.2) in is equivalent to the classical problem of finding all perfect squares and perfect cubes with given difference, which goes back at least to Bachet 1621. Mordell [Mor22, Mor23], Faltings [Fal83b] and Kim [Kim10] showed finiteness of (1.2) by using completely different proofs, and the first effective result for Mordell’s equation was provided by Baker [Bak68b]; see Section 7.3.1 where we briefly discuss methods which show finiteness of (1.2). On working out explicitly the method of this paper for the moduli schemes corresponding to Mordell equations, we get a new effective finiteness proof for (1.2). More precisely, if with the product taken over all rational primes with , then Corollary 7.4 proves that any solution of satisfies
This inequality allows in principle to find all solutions of any Mordell equation (1.2) and it provides in particular an entirely new proof of Baker’s classical result [Bak68b]. Moreover, the displayed estimate improves the actual best upper bounds for (1.2) in the literature and it refines and generalizes Stark’s theorem [Sta73]; see Section 7.3 for more details.
We observe that defines a canonical bijection between the set of finite sets of rational primes and the set of non-empty open subschemes of . In what follows in this paper (except Sections 7.2-7.4), we will adapt our notation to the algebraic geometry setting and the symbol will denote a base scheme.
1.1.2 Integral points on moduli schemes of elliptic curves
More generally, we now consider integral points on arbitrary moduli schemes of elliptic curves. We denote by and non-empty open subschemes of , with . Let be a moduli scheme of elliptic curves, which is defined over , and let be the maximal (possibly infinite) number of distinct level -structures on an arbitrary elliptic curve over ; see Section 3 for the definitions. We denote by the set of -points of the -scheme . Let be the pullback of the relative Faltings height by the canonical forget -map, defined in (3.3). Write with the product taken over all rational primes not in . We obtain in Theorem 7.1 the following result.
Theorem A. The following statements hold.
The cardinality of is at most with the product taken over all rational primes which divide .
If , then .
If the moduli problem is given with , then the explicit upper bound for the height in (ii) has the following application: In principle one can determine the abstract set up to a canonical bijection; see the discussion surrounding (3.3). Part (i) gives a quantitative finiteness result for provided that . In fact most moduli schemes of interest in arithmetic, in particular all explicit moduli schemes considered in this paper, trivially satisfy However, any scheme over an arbitrary -scheme is a moduli scheme of elliptic curves (see Section 3) and thus there exist many open subschemes and moduli schemes over such that is infinite.
In addition, we show that the Shimura-Taniyama conjecture := allows to deal with other classical Diophantine problems. For example, we consider cubic Thue equations, we derive an exponential version of Szpiro’s discriminant conjecture for any elliptic curve over , and we deduce an effective Shafarevich conjecture for elliptic curves over .
We remark that the theory of logarithmic forms gives more general versions of the results discussed so far, see [vK13, vK]. However, the approach via has other advantages. For instance, in the two examples which we worked out explicitly, we obtained upper bounds with numerical constants that are smaller than those coming from the theory of logarithmic forms. Furthermore, in the forthcoming joint work with Benjamin Matschke [vKM13], we will estimate more precisely the quantities appearing in our proofs to further improve our final numerical constants. This will allow us to practically resolve -unit and Mordell equations with “small” parameters. In fact the practical resolution of these Diophantine equations is still a challenging problem; see for example Gebel-Pethö-Zimmer [GPZ98] for partial results on Mordell’s equation. We also point out that has in addition the potential to find the solutions of Diophantine equations without using height bounds. For instance, we shall see in the proof of Theorem A that integral points on moduli schemes of elliptic curves correspond to elliptic curves over of bounded conductor, which in turn correspond by to certain newforms of bounded level and such newforms can be computed by Cremona [Cre97]. We refer to [vKM13] for details.
1.1.3 Principal ideas of Theorem A
We continue the notation of the previous section. Let be as above and suppose that is a moduli scheme over with . To describe our finiteness proofs for , we denote by the set of isomorphism classes of elliptic curves over . Forgetting the level structure induces a canonical map (see Lemma 3.1)
which has fibers of cardinality at most . Hence to show finiteness of , it suffices to control . This can be done in two steps: (a) Finiteness of up to isogenies and (b) finiteness of each isogeny class of . Mazur-Kenku [Ken82] implies (b), and (a) follows from [BCDT01] which provides an abelian variety over of controlled dimension such that the generic fiber of any is a quotient
This leads to Theorem A (i). To prove the explicit height bounds in Theorem A (ii), it suffices by (1.3) to control the relative Faltings height of (see Section 2). We first work out explicitly an estimate of Frey [Fre89] which relies on several non-trivial results, including [Ken82]: If is modular, then Frey estimates in terms of the modular degree of the newform associated with . The theory of modular forms allows to bound in terms of the level of , and says that is modular. Hence one obtains an estimate for in terms of , which then leads to Theorem A (ii).
To obtain upper bounds for heights on which are different to , it remains to work out height comparisons. In the two examples discussed above, one can do this explicitly by using explicit formulas for certain (Arakelov) invariants of elliptic curves.
We emphasize that the crucial ingredients for Theorem A are (1.3) and the “geometric” version (1.4) of which relies inter alia on the Tate conjecture [Fal83b]. The other tools, such as Frey’s estimate, the theory of modular forms and the isogeny results of Mazur-Kenku [Ken82], can be replaced by Arakelov theory and isogeny estimates; see Section 1.2.1 below. In fact the proof of Theorem A may be viewed as an application of a refined Arakelov-Faltings-Paršin method to moduli schemes of elliptic curves.
1.2 Effective Shafarevich conjecture
In 1983, Faltings [Fal83b] proved the Shafarevich conjecture [Sha62] for abelian varieties over number fields. It is known that an effective version of the Shafarevich conjecture would have striking Diophantine applications. For example, we show in Section 9 that the following effective Shafarevich conjecture implies the effective Mordell conjecture for any curve of genus at least 2, defined over an arbitrary number field.
Let be a non-empty open subscheme of , and let be an integer. We denote by the stable Faltings height, defined in Section 2.
Conjecture (). There exists an effective constant , depending only on and , such that any abelian scheme over of relative dimension satisfies
We mention that Conjecture is widely open if and we point out that implies in particular the “classical” effective Shafarevich conjecture for curves over arbitrary number fields; we refer to Section 9 for a discussion of Conjecture .
Let be an abelian scheme over of relative dimension . We say that is of -type if there exists a number field of degree together with an embedding Here denotes the ring of -group scheme morphisms from to . More generally, we say that is of product -type if is isogenous to a product of abelian schemes over of -type; see Section 8 for a discussion of abelian schemes of product -type. Write with the product taken over all rational primes not in . We prove in Theorem 9.2 the following result.
Theorem B. If is of product -type, then
This explicit Diophantine inequality establishes in particular the effective Shafarevich conjecture for all abelian schemes of product -type. In addition, we deduce in Corollary 9.4 new cases of the “classical” effective Shafarevich conjecture for curves, and we derive in Corollary 9.5 new isogeny estimates for any of product -type.
Next, we consider the set formed by the isomorphism classes of abelian schemes over of relative dimension which are of product -type. We obtain in Theorem 9.6 the following quantitative finiteness result for .
Theorem C. The cardinality of is at most
We deduce in Corollary 9.7 explicit finiteness results for -isogeny classes of abelian varieties over of product -type. Further, we mention that Brumer-Silverman [BS96], Poulakis [Pou00] and Helfgott-Venkatesh [HV06] established the important special case of Theorem C. They used completely different arguments which in fact give a better exponent for if . However, their methods crucially depend on the explicit nature of elliptic curves and they do not allow to deal with higher dimensional abelian varieties; see the discussion surrounding Proposition 6.4 for more details.
We remark that the above results open the way for the effective study of classes of Diophantine equations which appear to be beyond the reach of the known effective methods. For instance, Theorems B and C are the main tools of the joint paper with Arno Kret [vKK]. Therein we combine these results with canonical forgetful maps in the sense of (1.3), and we prove quantitative and effective finiteness results for integral points on higher dimensional moduli schemes which parametrize abelian schemes of -type. In particular, we work out the case of Hilbert modular varieties.
1.2.1 Principal ideas of Theorems B and C
We continue the notation of the previous section. Let and be as above, and let be the stable Faltings height. Suppose that is an abelian scheme over of relative dimension which is of product -type. Write for the generic fiber of . To prove Theorem B we combine ideas of Faltings [Fal83b] with the following tools:
Let be the conductor of , defined in Section 2.2. In the proof of Theorem B we first consider the case when is -simple. A result of Carayol [Car86] allows to control the number in (i). This together with (i)-(iii) leads to an effective bound for in terms of and , and then in terms of and since is an abelian scheme over . To reduce the general case of Theorem B to the case when is -simple, we use inter alia Poincaré’s reducibility theorem and the isogeny estimates in (ii).
We now describe the principal ideas of Theorem C. Following Faltings [Fal83b] we divide our quantitative finiteness proof for into two parts: (a) Finiteness of up to isogenies and (b) finiteness of each isogeny class of . To prove (a) we use (i) and we show that any -simple “factor” of is a quotient
where is an integer depending only on and . To show (b) we combine Theorem B with an estimate of Masser-Wüstholz [MW93, MW95] for the minimal degree of isogenies of abelian varieties which is based on transcendence theory. In fact we use here the most recent version of the Masser-Wüstholz estimate, due to Gaudron-Rémond [GR12].
We remark that results from transcendence theory are not crucial to prove Conjecture for abelian schemes over of product -type (resp. to effectively estimate ). However, they lead to an upper bound for (resp. for ) which is exponentially (resp. double exponentially) better in terms of and , than the estimate which would follow by using the results in (ii) based on Faltings’ method.
1.3 Plan of the paper
In Section 2 we discuss properties of Faltings heights and of the conductor of abelian varieties over number fields. In Section 3 we give Paršin constructions for moduli schemes of elliptic curves and in Section 4 we collect results which control the variation of Faltings heights in an isogeny class. In Section 5 we use the theory of modular forms to bound the modular degree of elliptic curves over . We also estimate the stable Faltings heights of certain classical modular Jacobians. Then we prove in Section 6 an explicit height conductor inequality for elliptic curves over and we derive some applications. In Section 7 we give our effective finiteness results for integral points on moduli schemes of elliptic curves. In Section 8 we prove a height conductor inequality for abelian varieties over of product -type. Finally, we establish in Section 9 the effective Shafarevich conjecture for abelian schemes of product -type and we deduce some applications.
I would like to thank Richard Taylor for answering several questions, in particular for proposing a first strategy to prove Lemma 5.1. Many thanks go to Bao le Hung, Arno Kret, Benjamin Matschke, Richard Taylor, Jack Thorne and Chenyan Wu for motivating discussions. Parts of the results were obtained when I was a member (2011/12) at the IAS Princeton, supported by the NSF under agreement No. DMS-0635607. I am grateful to the IAS and the IHÉS for providing excellent working conditions. Also, I would like to apologize for the long delay between the first presentation (2011) of the initial results on -unit and Mordell equations and the completion (2013) of the manuscript. The delay resulted from the attempt to understand the initial examples in a way which is more conceptual and which is suitable for generalizations.
1.5 Conventions and notations
We identify a nonzero prime ideal of the ring of integers of a number field with the corresponding finite place of and vice versa. We write for the number of elements in the residue field of , we denote by the order of in a fractional ideal of and we write (resp. ) if (resp. ). If is an abelian variety over with semi-stable reduction at all finite places of , then we say that is semi-stable.
Let be an arbitrary scheme. We often identity an affine scheme with the ring . If and are -schemes, then we denote by the set of -scheme morphisms from to and we write for the base change of from to . Further, if and are abelian schemes over , then we denote by the abelian group of -group scheme morphisms from to and we write for the endomorphism ring of . Following [BLR90], we say that is a Dedekind scheme if is a normal noetherian scheme of dimension 0 or 1.
By we mean the principal value of the natural logarithm and we define the maximum of the empty set and the product taken over the empty set as . For any set , we denote by the (possibly infinite) number of distinct elements of . Let be real valued functions on . We write if there exists a constant such that . Finally, for any map , we write if for all there exists a constant , depending only on , such that
2 Height and conductor of abelian varieties
Let be a number field and let be an abelian variety over . In the first part of this section, we recall the definition of the relative and the stable Faltings height of , and we review fundamental properties of these heights. In the second part, we define the conductor of and we recall useful properties of .
2.1 Faltings heights
We begin to define the relative and stable Faltings height of following [Fal83b, p.354]. If then we set . We now assume that has positive dimension . Let be the spectrum of the ring of integers of . We denote by the Néron model of over , with zero section . Let be the sheaf of relative differential -forms of . We now metrize the line bundle on . For any embedding , we denote by the base change of to with respect to . We choose a nonzero global section of . Let be the positive real number that satisfies
where denotes the holomorphic differential form on which is induced by . Then the relative Faltings height of is the real number defined by
with the sum taken over all embeddings . The product formula assures that this definition does not depend on the choice of . The relative Faltings height is compatible with products of abelian varieties: If is an abelian variety over , then To see the behaviour of under base change we take a finite field extension of . The universal property of Néron models implies
This inequality can be strict and thus the height is in general not stable under base change. To obtain a stable height we may (see [GR72]) and do take a finite extension of such that is semi-stable. The stable Faltings height of is defined as
This definition does not depend on the choice of , since the formation of the identity components of the corresponding semi-stable Néron models commutes with the induced base change. In particular, inequality (2.1) becomes an equality when is replaced by . Further, we define . We shall need an effective lower bound for in terms of the dimension of . An explicit result of Bost [Bos96a] gives
We shall state several of our results in terms of or and therefore we now briefly discuss important differences between these heights. From (2.1) we deduce that Further, as already observed, the height has the advantage over that it is stable under base change. On the other hand, has in general weaker finiteness properties. For instance, there are only finitely many -isomorphism classes of elliptic curves over of bounded , while is bounded on the infinite set given by the -isomorphism classes of elliptic curves of any fixed -invariant in .
More generally, let be a connected Dedekind scheme with field of fractions . If is an abelian scheme over , then we define the stable and relative Faltings height of by and respectively. Here is the generic fiber of .
We first define the conductor of an arbitrary abelian variety over any number field . Let be a finite place of . We denote by the usual conductor exponent of at , see for example [Ser70, Section 2.1] for a definition. The conductor of is defined by
with the product taken over all finite places of . In particular, and . We now recall some useful properties of and . It holds that if and only if has good reduction at . Furthermore, if is an abelian variety over which is -isogenous to , then and thus . Finally, if is an abelian variety over and if , then and hence .
We shall need an explicit upper bound for in terms of and . Brumer-Kramer [BK94] obtained such a bound by refining earlier work of Serre [Ser87, Section 4.9] and of Lockhart-Rosen-Silverman [LRS93]. To state the main result of [BK94] we have to introduce some notation. Let be the residue characteristic of , let be the ramification index of , and let be the largest integer that satisfies . We define for the -adic expansion of with integers . Then [BK94, Theorem 6.2] gives
More generally, if is a connected Dedekind scheme with field of fractions and if is an abelian scheme over , then we define the conductor of by .
3 Paršin constructions: Forgetting the level structure
Paršin [Par68] discovered a link between the Mordell and the Shafarevich conjecture which is now commonly known as Paršin construction or Paršin trick. This link gives a finite map from the set of rational points of into the integral points of a certain moduli space, where is a curve of genus at least two which is defined over a number field.
In the first part of this section, we use the moduli problem formalism to obtain tautological Paršin constructions for moduli schemes of elliptic curves. In the second part, we explicitly work out this idea for and once punctured Mordell elliptic curves. This results in completely explicit Paršin constructions for these hyperbolic curves.
3.1 Moduli schemes
We begin to introduce some notation and terminology. Let be an arbitrary scheme. An elliptic curve over is an abelian scheme over of relative dimension one. A morphism of elliptic curves over is a morphism of abelian schemes over . We denote by
the set of isomorphism classes of elliptic curves over . On following Katz-Mazur [KM85, p.107], we write for the category of elliptic curves over variable base-schemes: The objects are elliptic curves over schemes and the morphisms are given by cartesian squares of elliptic curves. Let be the category of sets and let be a contravariant functor from to . We say that is a moduli problem on and we define
with the supremum taken over all elliptic curves over . In other words, is the maximal (possibly infinite) number of distinct level -structures on an arbitrary elliptic curve over . A scheme is called a moduli scheme (of elliptic curves) if there exists a moduli problem on which is representable by an elliptic curve over . The following lemma may be viewed as a tautological Paršin construction for moduli schemes.
Suppose is a moduli scheme, defined over a scheme . If is a -scheme, then there is a map with fibers of cardinality at most .
We notice that the statement is intuitively clear, since is essentially the set of elliptic curves over with “level -structure” and the map is essentially “forgetting the level -structure”. We now verify that this intuition is correct.
By assumption, there exists a contravariant functor from to which is representable by an elliptic curve over . Suppose and are elliptic curves over a scheme , with and . Then the pairs and are called isomorphic if there exists an isomorphism of objects in with . Let be the set of isomorphism classes of such pairs . Then defines a contravariant functor from the category of schemes to , which is representable by since is representable by an elliptic curve over . Thus we obtain an inclusion , which composed with
gives a map . Suppose is the fiber of this map over a point in . Then all are isomorphic objects of . Therefore, after applying suitable isomorphisms of objects in , we may and do assume that all coincide. This shows that and then we conclude Lemma 3.1. ∎
We call the map constructed in Lemma 3.1 the forget -map. To discuss some fairly general examples of moduli schemes we consider an arbitrary scheme . If there exists an elliptic curve over , then is a moduli scheme with . This shows in particular that any -scheme is a moduli scheme, since there exists an elliptic curve over and the base change is an elliptic curve over . Next, we discuss a classical example of a moduli problem. Let be an integer and consider the “naive” level moduli problem from to , defined by
Here we view as a constant -group-scheme and is the kernel of the -homomorphism “multiplication by ” on the elliptic curve over . If is non-empty and if is connected, then we explicitly compute
If then [KM85, Corollary 4.7.2] gives that is a representable moduli problem on , with moduli scheme a smooth affine curve over .
3.2 Explicit constructions
We introduce and recall some notation. Let be a number field and write for the spectrum of the ring of integers of . In the remaining of this section, we denote by
either a non-empty open subscheme of or the spectrum of the function field of and we write . Let be an elliptic curve over . We denote by and by the relative and the stable Faltings height of the generic fiber of respectively, see Section 2 for the definitions. Let be the conductor of defined in Section 2.2 and let be the norm from to of the usual minimal discriminant ideal of over . We observe that and define real valued functions on .
Let be a moduli scheme defined over , let be a non-empty open subscheme of and let be the forget -map from Lemma 3.1. On pulling back the relative Faltings height by , we get a height on defined by
The height has the following properties: If , then Lemma 3.1 together with Lemma 3.5 below shows that there exist only finitely many with bounded. Furthermore, if is given with , then the proof of Lemma 3.1 together with Lemma 3.5 below implies that one can in principle determine, up to a canonical bijection, the set of points with effectively bounded.
Let be the absolute value of the discriminant of over , let be the degree of over and let be the cardinality of the class group of . We define
with the product taken over all ; notice that if . Further, we say that any nonzero is invertible on if and are both in . For any vector with coefficients in , we denote by the usual absolute logarithmic Weil height of which is defined in [BG06, 1.5.6].
3.2.1 -unit equations
We continue the notation introduced above and we now give an explicit Paršin construction for “-unit equations”. The solutions of such equations correspond to -points of
To simplify notation we write . For any , we define .222If is an affine -scheme, and , then denotes the image of under the ring morphism which corresponds to . We say that a map of sets is finite if all its fibers are finite.
Suppose that is an open subscheme of , with invertible on . Then there exists a finite map with the following properties.
Suppose and . Then it holds and
There is an elliptic curve over that satisfies and
If has trivial class group, then extends to an elliptic curve over
and . If , then
In this article, we shall use Proposition 3.2 only for one dimensional and . However, the height inequalities obtained in this proposition may be also of interest for . We mention that the number 6 in these height inequalities is optimal.
To prove Proposition 3.2 we shall use inter alia the following lemma.
If is an elliptic curve over , then
For any embedding , we take such that the base change of to with respect to takes the form and such that . We write and . From [Sil86, Proposition 1.1] we get
with the sum taken over all embeddings . Here denotes the complex absolute value. Further, on using the elementary inequalities and , we deduce the estimate
This together with the displayed formula for implies the statement. ∎
We remark that the proof shows in addition that Faltings’ delta invariant [Fal84, p.402] of a compact connected Riemann surface of genus one satisfies
Indeed, this follows directly from (3.4) and Faltings’ explicit formula [Fal84, Lemma c), p.417] for . We mention that it is an important open problem to obtain explicit lower bounds, in terms of the genus, for the Faltings delta invariant of compact connected Riemann surfaces of arbitrary positive genus.
We shall need an estimate for the conductor. If is a closed point of and if denotes the conductor exponent at of an elliptic curve over (see Section 2.2), then
This follows directly from the result of Brumer-Kramer which we stated in (2.4).
Proof of Proposition 3.2.
We observe that if is empty, then all statements are trivial. Hence we may and do assume that is not empty. We denote by the spectrum of for an “indeterminate”. Then we observe that
defines an (universal) elliptic curve over . We take . On using that , we obtain a morphism induced by . Let be the fiber product of with this morphism . Then is an elliptic curve over and therefore we see that
defines a map If satisfies , then it follows that with pairwise distinct . Thus is finite.
We now prove (i). In what follows we write for to simplify notation. The -invariant of the generic fiber of satisfies
This implies that for any finite place of with and that for any embedding with , where denotes the complex absolute value. We deduce
This implies an upper bound for as stated in (i). Next, we prove the claimed estimate for the conductor of . This estimate holds trivially if , and we now assume that . In what follows we denote by a closed point of . Let be the conductor exponent of at . If , then has good reduction at , since is smooth and projective, and we obtain . Thus the estimates in (3.5) for if combined with if lead to an upper bound for as stated in (i).
To show (ii) we observe that the statement is trivial if . Hence we may and do assume that . As in the proof of [vK12, Lemma 4.1], we see that Minkowski’s theorem gives an open subscheme of with the following properties. There are at most points in , any satisfies and the class group of is trivial. Then we may and do take coprime elements such that
Let be an elliptic curve over defined by the Weierstrass equation . We observe that is geometrically isomorphic to . This implies that the -invariant of coincides with and . We now prove the claimed estimate for the conductor of . Let and be the usual quantities associated to the above Weierstrass equation of , see [Sil09, p.42]. They take the form
Let be the conductor exponent of at . First, we assume that with . If , then it follows that , since are coprime and . This implies that the above Weierstrass equation is minimal at and then [Sil09, p.196] proves that is semi-stable at . We conclude . Next, we assume . In the proof of (i) we showed . This implies that , since is geometrically isomorphic to and is semi-stable at . On combining the above observations, we deduce
with the product taken over all such that or . Therefore, on using the properties of , we see that the estimates in (3.5) for if combined with if imply an upper bound for as claimed in (ii).
It remains to prove (iii). We notice that the first assertion of (iii) is trivial if . If has trivial class group, then we can take in the proof of (ii): It follows that for any closed point and that
with the product taken over all with . This shows that is the generic fiber of an elliptic curve over and that . If , then we obtain that , and [Sil09, p.257] shows that . Therefore Lemma 3.3 proves (iii). This completes the proof of Proposition 3.2.∎
We remark that the elliptic curve over , which appears in the above proof, represents the moduli problem on defined in [KM85, p.111]. The moduli scheme is defined over . If is invertible on and if , then it follows that and that the map in Proposition 3.2 coincides with the map in Lemma 3.1. However, to get our explicit inequalities in Proposition 3.2 it is necessary to take into account the particular shape of .
3.2.2 Mordell equations
We continue the notation introduced above and we now give an explicit Paršin construction for Mordell equations. For any nonzero , we obtain that
defines an affine Mordell curve over . To state our next result we have to introduce some additional notation. If then we write . Let be the regulator of and let be the rank of the free part of the group of units of . We define
and we observe that when . The origin of the constant shall be explained below Lemma 3.5. To measure the number , we use inter alia the quantity
with the product taken over all closed points with . We observe that and if , then for the norm from to .
Suppose that is an open subscheme of , with invertible on . Then there is a map with the following properties.
The map is finite. Furthermore, if are the only 12th roots of unity
in , then is injective.
Suppose and . Then it holds and
If, in addition, , then
To prove Proposition 3.4 we shall use a lemma which relates heights of elliptic curves. We recall that denotes the generic fiber of an elliptic curve over . Let be a Weierstrass model of over with discriminant , see for example [Liu02, Section 9.4.4] for a definition of and . To measure we take the height
where and are the usual quantities of a defining Weierstrass equation of , see [Sil09, p.42]. It turns out that the definition of does not depend on the choice of the defining Weierstrass equation of . We obtain the following lemma.
Suppose that is an elliptic curve over . Then there exists a Weierstrass model of over that satisfies
If , then this lemma would follow on calculating the constants in Silverman’s [Sil86, Proposition 2.1, Corollary 2.3]. However, the proof of [Sil86, Corollary 2.3] does not generalize directly to arbitrary , since it uses that the ring of integers of has class number one and unit group . To deal with arbitrary we apply a classical theorem of Minkowski and a result which is based on estimates for certain fundamental units of . This leads to a dependence of the constant on , and on , , .
Proof of Lemma 3.5.
On combining [Sil09, p.264] with a classical result of Minkowski, we obtain a Weierstrass model of over of discriminant such that
for the minimal discriminant ideal of and an ideal with . For any nonzero , an application of [GY06, Lemma 3]333This result relies on estimates for certain fundamental units of . with gives such that . Hence, on using (3.8), we obtain a defining Weierstrass equation of , with quantities and discriminant , such that